Statistics Chapter 3: Central Tendency

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Questions and Answers

What is the median of the following data set: 10, 2, 7, 14, 5, 9?

8.5

7.5

8 (correct)

Given the dataset: 3, 6, 6, 8, 10, 12. Which of the following statements is true?

The mean is equal to the mode

The median is equal to the mean

The mode is greater than the median (correct)

The median is greater than the mean

Which of the following best describes a disadvantage of using the median as a measure of central tendency?

It is significantly affected by extreme scores.

It requires more computational steps than the mean.

It may not accurately represent the typical value. (correct)

It cannot be applied to categorical data.

Which of the following datasets would have both a mode and a median equal to 9?

7, 8, 9, 9, 10, 11 (C) Signup and view all the answers

In a dataset, the mean is $250, the median is $200 and the mode is $150. Which measure is most resistant to extreme values?

The median (A) Signup and view all the answers

Which measure of central tendency is most susceptible to misrepresenting the center of a distribution due to extreme scores?

Mean (C) Signup and view all the answers

If a dataset has an even number of values, how is the median calculated?

The average of the two middle numbers is derived. (C) Signup and view all the answers

What does the symbol μ, represent in the context of central tendency calculations?

Population mean (C) Signup and view all the answers

Which measure of central tendency provides the most stable estimate of the population mean across repeated samples?

Mean. (B) Signup and view all the answers

Given the set of numbers: 2, 8, 5, 2, 6, what is the mode?

2 (B) Signup and view all the answers

What action is required before calculating the median of a dataset?

List the numbers in ascending order (B) Signup and view all the answers

In the distribution: 3, 3, 6, 8, 9, which value represents the median?

6 (B) Signup and view all the answers

A sample of values is: 10, 20, 30, and 40. What is the sample mean?

25 (B) Signup and view all the answers

Which of the following is an advantage of using the mode as a measure of central tendency?

It can be applied to nominal data. (A) Signup and view all the answers

What does a positively skewed distribution indicate?

The distribution trails off to the right. (C) Signup and view all the answers

In a skewed distribution, where is the median typically located?

Typically but not always between the mean and the mode. (C) Signup and view all the answers

What is a key characteristic of a normal distribution?

Data are symmetrically distributed around the mean, median, and mode. (C) Signup and view all the answers

Which of the following is true about a perfectly normal distribution?

It is perfectly symmetrical and unimodal. (C) Signup and view all the answers

Which measure of central tendency is most appropriate for normally distributed data?

The mean. (D) Signup and view all the answers

Which measure of central tendency is most appropriate for modal distributions?

The mode. (A) Signup and view all the answers

What does the concept of 'dispersion' refer to in a distribution?

How scores are spread out on the x-axis. (A) Signup and view all the answers

What does the range of a dataset represent?

The difference between the highest and lowest scores. (D) Signup and view all the answers

Why might the range be less informative for a dataset containing outliers?

Because outliers significantly inflate the range, creating a misleading representation of score distribution. (C) Signup and view all the answers

What is variance?

The average squared distance that scores deviate from their mean. (B) Signup and view all the answers

What is the sum of squares (SS)?

The sum of the squared deviations of scores from their mean. (B) Signup and view all the answers

In the formula for sample variance, why is the sum of squares (SS) divided by N-1, instead of N?

To account for the degrees of freedom in the sample. (C) Signup and view all the answers

What are degrees of freedom (df)?

The number of independent pieces of information that are free to vary, minus the number of mathematical restrictions. (C) Signup and view all the answers

If three numbers add up to 20, and two of the numbers are 5 and 8, how many degrees of freedom are there?

2 (B) Signup and view all the answers

Given the following numbers: 2, 4, 6, and 8, what is the range?

6 (A) Signup and view all the answers

What is the relationship between variance and standard deviation?

Standard deviation is the square root of the variance. (B) Signup and view all the answers

Why are computational formulas preferred over definitional formulas for calculating variance?

Definitional formulas are more prone to rounding errors. (C) Signup and view all the answers

The standard deviation is described as:

the average distance that scores deviate from their mean (C) Signup and view all the answers

In the provided data, what does a higher rating indicate?

More attractiveness (C) Signup and view all the answers

How many scores are there in Set 4 based on the provided information?

8 (B) Signup and view all the answers

What is the purpose of calculating the standard deviation?

To measure the dispersion of scores around the mean (A) Signup and view all the answers

According to the given information, what is the standard deviation of Set 4?

1.69 (A) Signup and view all the answers

Which of the following formulas represents the sample standard deviation?

$ \sqrt{\frac{\sum x - \frac{(\sum x)^2}{N}}{N-1}}$ (C) Signup and view all the answers

Study Notes

Chapter 3: Central Tendency

Central tendency measures the center of a distribution. Examples include mean, median, and mode.
Scientists, therapists, and educators frequently need to understand the central tendency of data. For example, finding the average number of symptoms in a disorder, or the most frequent symptom.

Mean

The mean is the sum of scores divided by the number of scores. This is also known as the average.
Population mean (μ): μ = Σx/N where μ is the Greek letter mu, Σx is the sum of all scores, and N is the population size.
Sample mean (M): M = Σx/N where N is the size of the sample.
The mean may be misleading if there are extreme values (outliers) in the data. For example, a psychotherapy school might claim a mean hourly rate of $500, but if a few psychotherapists charge $2,100/hour and the others are in the $100 range, the average will be inflated. The median will be more accurate in cases like this.

Median

The median is the middle number in an ordered set of numbers.
For odd-sized sets, it is the middle value after ordering the data. For even-sized sets it is the average of the two middle values.
Example: Find the median of the odd set 1, 0, 5, 4, 6; Order the numbers: 0, 1, 4, 5, 6; The middle number (3rd number) is 4.
Example: Find the median of the even set 2, 8, 0, 6, 4, 5; Order the numbers: 0, 2, 4, 5, 6, 8; The middle two numbers are 4 and 5; The average of these two numbers is 4.5.
The median is less sensitive to outliers than the mean.

Mode

The mode is the most frequently occurring score or value.
Example: Find the mode for the data 1, 2, 2, 2, 3, 4. The mode is 2.
Example: Find the mode for the data 1, 2, 2, 3, 4, 4. The modes are 2 and 4.
Example: Find the mode for the data blue, blue, pink, pink, gray, gray, gray. The mode is gray.
The mode can only be determined from nominal scale data.

Describing Distributions

Graphed distributions can vary in skew (symmetry) and kurtosis (pointedness).
Positively skewed distribution: the tail of the distribution trails off to the right.
Negatively skewed distribution: the tail of the distribution trails off to the left.
Normal distribution: a symmetrical distribution where the mean, median, and mode are all located at the center.

The Empirical Rule

For normally distributed data:
- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.

Chapter 4: Variability

Variability describes how dispersed or spread out data points are.
Range: the difference between the highest and lowest score. The range is helpful if there are no major outliers.
Variance: the average squared distance that scores deviate from their mean.
Standard Deviation: the average distance that scores deviate from their mean. (The standard deviation is the square root of the variance).
The Empirical Rule is used to understand the proportion of data points falling within specific ranges of the mean using the standard deviation.

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Explore the measures of central tendency including mean, median, and mode. Understand how these statistics are calculated and applied in various fields like science and psychology. This quiz will help solidify your knowledge of how to interpret data effectively.